Tissue add-on: How to use the Advanced Settings for animation?
The Tessellate Tool's Advanced Settings are grayed out unless the "base object" has a Vertex Group (with vertices assigned to it).

So that answers the original question. But let's see what these Advanced Settings might actually be used for in practice, and go through the steps.

A few more points that aren't obvious:


The Active Shape Key is the one that will be used.
Shape Key slider values will not be taken into account.
If "Map Vertex Group" is not enabled, the "Use Shape Keys" option will fail.
Shape Key data is not preserved as such. The tessellation result is just a mesh in that single shape.
To get the animation back we need to create multiple tessellations and use the Join As Shapes operation to recombine them as one mesh.


Here's an animated mini-tutorial: 



And a sample result: 



Go nuts...